url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | have : ¬Eval (Model.structure T Γ) ![u] Semiterm.fvar p := by
simpa[eval_substs, Matrix.constant_eq_singleton] using semanticMainLemma_val ([→ u].hom p) hu | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
u : Semiterm L ℕ 0
h... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
u : Semiterm L ℕ 0
h... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | simp | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
u : Semiterm L ℕ 0
h... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
u : Semiterm L ℕ 0
h... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | exact ⟨u, this⟩ | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
u : Semiterm L ℕ 0
h... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | simpa[eval_substs, Matrix.constant_eq_singleton] using semanticMainLemma_val ([→ u].hom p) hu | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
u : Semiterm L ℕ 0
hu : [→ u].h... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | simp | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
⊢ ¬(Evalf (Model.structure T Γ)... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
⊢ ∀ (x : Model T Γ), ¬(Eval (Mo... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | intro u | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
⊢ ∀ (x : Model T Γ), ¬(Eval (Mo... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
u : Model T Γ
⊢ ¬(Eval (Model.s... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | have : [→ u].hom p ∈ ⛓️ := chainSet_ex nwf h u | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
u : Model T Γ
⊢ ¬(Eval (Model.s... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
u : Model T Γ
this : [→ u].hom ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | have : ¬Eval (Model.structure T Γ) ![u] Semiterm.fvar p := by
simpa[eval_substs, Matrix.constant_eq_singleton] using semanticMainLemma_val ([→ u].hom p) this | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
u : Model T Γ
this : [→ u].hom ... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
u : Model T Γ
this✝ : [→ u].hom... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | assumption | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
u : Model T Γ
this✝ : [→ u].hom... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | simpa[eval_substs, Matrix.constant_eq_singleton] using semanticMainLemma_val ([→ u].hom p) this | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∃' p ∈ ⛓️
u : Model T Γ
this : [→ u].hom ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.Model.models | [314, 1] | [315, 80] | intro σ hσ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
⊢ ∀ ⦃f : Sentence L⦄, f ∈ T → (structure T Γ).toStruc ⊧ f | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
hσ : σ ∈ T
⊢ (structure T Γ).toStruc ⊧ σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.Model.models | [314, 1] | [315, 80] | simpa using semanticMainLemma_val nwf _ (chainSet_id nwf hσ) | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
hσ : σ ∈ T
⊢ (structure T Γ).toStruc ⊧ σ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemmaTop | [317, 1] | [318, 82] | simp[chain, chainU, h] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticFormula L
h : p ∈ Γ
⊢ p ∈ {x | x ∈ ⛓️[0]} | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.Frame.terminal.rel | [50, 1] | [51, 93] | aesop | x y : terminal.World
⊢ x ≺ y ↔ x = y | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.Frame.terminal.relItr | [53, 1] | [57, 41] | induction n with
| zero => simp;
| succ n ih => simp; use x; simp [ih]; | n : ℕ
x y : terminal.World
⊢ x ≺^[n] y ↔ x = y | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.Frame.terminal.relItr | [53, 1] | [57, 41] | simp | case zero
x y : terminal.World
⊢ x ≺^[0] y ↔ x = y | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.Frame.terminal.relItr | [53, 1] | [57, 41] | simp | case succ
x y : terminal.World
n : ℕ
ih : x ≺^[n] y ↔ x = y
⊢ x ≺^[n + 1] y ↔ x = y | case succ
x y : terminal.World
n : ℕ
ih : x ≺^[n] y ↔ x = y
⊢ ∃ x, RelItr (fun x x_1 => x ≺ x_1) n x y |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.Frame.terminal.relItr | [53, 1] | [57, 41] | use x | case succ
x y : terminal.World
n : ℕ
ih : x ≺^[n] y ↔ x = y
⊢ ∃ x, RelItr (fun x x_1 => x ≺ x_1) n x y | case h
x y : terminal.World
n : ℕ
ih : x ≺^[n] y ↔ x = y
⊢ RelItr (fun x x_1 => x ≺ x_1) n x y |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.Frame.terminal.relItr | [53, 1] | [57, 41] | simp [ih] | case h
x y : terminal.World
n : ℕ
ih : x ≺^[n] y ↔ x = y
⊢ RelItr (fun x x_1 => x ≺ x_1) n x y | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.atom_def | [125, 1] | [125, 78] | simp [Satisfies] | α : Type u_1
M : Model α
w : M.World
p q : Formula α
a : α
⊢ w ⊧ atom a ↔ M.Valuation w a | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.top_def | [126, 1] | [126, 62] | simp [Satisfies] | α : Type u_1
M : Model α
w : M.World
p q : Formula α
⊢ Satisfies M w ⊤ ↔ True | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.bot_def | [127, 1] | [127, 63] | simp [Satisfies] | α : Type u_1
M : Model α
w : M.World
p q : Formula α
⊢ Satisfies M w ⊥ ↔ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.and_def | [128, 1] | [128, 75] | simp [Satisfies] | α : Type u_1
M : Model α
w : M.World
p q : Formula α
⊢ Satisfies M w (p ⋏ q) ↔ Satisfies M w p ∧ Satisfies M w q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.or_def | [129, 1] | [129, 75] | simp [Satisfies] | α : Type u_1
M : Model α
w : M.World
p q : Formula α
⊢ Satisfies M w (p ⋎ q) ↔ Satisfies M w p ∨ Satisfies M w q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.imp_def | [130, 1] | [130, 91] | simp [Satisfies, imp_iff_not_or] | α : Type u_1
M : Model α
w : M.World
p q : Formula α
⊢ Satisfies M w (p ⟶ q) ↔ Satisfies M w p → Satisfies M w q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.not_def | [131, 1] | [131, 65] | simp [Satisfies] | α : Type u_1
M : Model α
w : M.World
p q : Formula α
⊢ Satisfies M w (~p) ↔ ¬Satisfies M w p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.box_def | [132, 1] | [132, 80] | simp [Satisfies] | α : Type u_1
M : Model α
w : M.World
p q : Formula α
⊢ Satisfies M w (□p) ↔ ∀ (w' : M.Frame.World), w ≺ w' → Satisfies M w' p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.dia_def | [133, 1] | [133, 80] | simp [Satisfies] | α : Type u_1
M : Model α
w : M.World
p q : Formula α
⊢ Satisfies M w (◇p) ↔ ∃ w', w ≺ w' ∧ Satisfies M w' p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | induction n generalizing w with
| zero => simp;
| succ n ih =>
constructor;
. intro h w' hww';
simp at h;
obtain ⟨v, hwv, hvw'⟩ := hww';
exact (ih.mp $ h _ hwv) w' hvw';
. simp;
intro h w' hww';
apply ih.mpr;
intro v hwv;
exact h v w' hww' hwv; | α : Type u_1
M : Model α
w : M.World
p q : Formula α
n : ℕ
⊢ Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | simp | case zero
α : Type u_1
M : Model α
p q : Formula α
w : M.World
⊢ Satisfies M w (□^[0]p) ↔ ∀ (v : M.Frame.World), w ≺^[0] v → Satisfies M v p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | constructor | case succ
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
⊢ Satisfies M w (□^[(n + 1)]p) ↔ ∀ (v : M.Frame.World), w ≺^[n + 1] v → Satisfies M v p | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
⊢ Satisfies M w (□^[(n + 1)]p) → ∀ (v : M.Frame.World), w ≺^[n + 1] v → Satisfies M v p
case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | . intro h w' hww';
simp at h;
obtain ⟨v, hwv, hvw'⟩ := hww';
exact (ih.mp $ h _ hwv) w' hvw'; | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
⊢ Satisfies M w (□^[(n + 1)]p) → ∀ (v : M.Frame.World), w ≺^[n + 1] v → Satisfies M v p
case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
... | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
⊢ (∀ (v : M.Frame.World), w ≺^[n + 1] v → Satisfies M v p) → Satisfies M w (□^[(n + 1)]p) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | . simp;
intro h w' hww';
apply ih.mpr;
intro v hwv;
exact h v w' hww' hwv; | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
⊢ (∀ (v : M.Frame.World), w ≺^[n + 1] v → Satisfies M v p) → Satisfies M w (□^[(n + 1)]p) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | intro h w' hww' | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
⊢ Satisfies M w (□^[(n + 1)]p) → ∀ (v : M.Frame.World), w ≺^[n + 1] v → Satisfies M v p | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
h : Satisfies M w (□^[(n + 1)]p)
w' : M.Frame.World
hww' : w ≺^[n + 1] w'
⊢ Satisfies M w' p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | simp at h | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
h : Satisfies M w (□^[(n + 1)]p)
w' : M.Frame.World
hww' : w ≺^[n + 1] w'
⊢ Satisfies M w' p | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
w' : M.Frame.World
hww' : w ≺^[n + 1] w'
h : ∀ (w' : M.Frame.World), w ≺ w' → Satisfies M w' ((UnaryModalOperator.mop true)^[n] p)
⊢ Satisfies M w' p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | obtain ⟨v, hwv, hvw'⟩ := hww' | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
w' : M.Frame.World
hww' : w ≺^[n + 1] w'
h : ∀ (w' : M.Frame.World), w ≺ w' → Satisfies M w' ((UnaryModalOperator.mop true)^[n] p)
⊢ Satisfies M w' p | case succ.mp.intro.intro
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
w' : M.Frame.World
h : ∀ (w' : M.Frame.World), w ≺ w' → Satisfies M w' ((UnaryModalOperator.mop true)^[n] p)
v : M.Frame.World
hwv : w ≺ v... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | exact (ih.mp $ h _ hwv) w' hvw' | case succ.mp.intro.intro
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
w' : M.Frame.World
h : ∀ (w' : M.Frame.World), w ≺ w' → Satisfies M w' ((UnaryModalOperator.mop true)^[n] p)
v : M.Frame.World
hwv : w ≺ v... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | simp | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
⊢ (∀ (v : M.Frame.World), w ≺^[n + 1] v → Satisfies M v p) → Satisfies M w (□^[(n + 1)]p) | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
⊢ (∀ (v x : M.Frame.World), w ≺ x → RelItr (fun x x_1 => x ≺ x_1) n x v → Satisfies M v p) →
∀ (w' : M.Frame.World), w ≺ w' → Satisfies M w' ((Un... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | intro h w' hww' | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
⊢ (∀ (v x : M.Frame.World), w ≺ x → RelItr (fun x x_1 => x ≺ x_1) n x v → Satisfies M v p) →
∀ (w' : M.Frame.World), w ≺ w' → Satisfies M w' ((Un... | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
h : ∀ (v x : M.Frame.World), w ≺ x → RelItr (fun x x_1 => x ≺ x_1) n x v → Satisfies M v p
w' : M.Frame.World
hww' : w ≺ w'
⊢ Satisfies M w' ((UnaryM... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | apply ih.mpr | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
h : ∀ (v x : M.Frame.World), w ≺ x → RelItr (fun x x_1 => x ≺ x_1) n x v → Satisfies M v p
w' : M.Frame.World
hww' : w ≺ w'
⊢ Satisfies M w' ((UnaryM... | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
h : ∀ (v x : M.Frame.World), w ≺ x → RelItr (fun x x_1 => x ≺ x_1) n x v → Satisfies M v p
w' : M.Frame.World
hww' : w ≺ w'
⊢ ∀ (v : M.Frame.World), ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | intro v hwv | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
h : ∀ (v x : M.Frame.World), w ≺ x → RelItr (fun x x_1 => x ≺ x_1) n x v → Satisfies M v p
w' : M.Frame.World
hww' : w ≺ w'
⊢ ∀ (v : M.Frame.World), ... | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
h : ∀ (v x : M.Frame.World), w ≺ x → RelItr (fun x x_1 => x ≺ x_1) n x v → Satisfies M v p
w' : M.Frame.World
hww' : w ≺ w'
v : M.Frame.World
hwv : w... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multibox_def | [135, 1] | [149, 29] | exact h v w' hww' hwv | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (□^[n]p) ↔ ∀ (v : M.Frame.World), w ≺^[n] v → Satisfies M v p
w : M.World
h : ∀ (v x : M.Frame.World), w ≺ x → RelItr (fun x x_1 => x ≺ x_1) n x v → Satisfies M v p
w' : M.Frame.World
hww' : w ≺ w'
v : M.Frame.World
hwv : w... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | induction n generalizing w with
| zero => simp;
| succ n ih =>
constructor;
. intro h;
obtain ⟨v, hwv, hv⟩ := by simpa using h;
obtain ⟨x, hvx, hx⟩ := ih.mp hv;
existsi x;
constructor;
. existsi v; simp_all;
. simpa;
. simp;
intro x y hwy hyx hx;
existsi y;
constructor;
. s... | α : Type u_1
M : Model α
w : M.World
p q : Formula α
n : ℕ
⊢ Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | simp | case zero
α : Type u_1
M : Model α
p q : Formula α
w : M.World
⊢ Satisfies M w (◇^[0]p) ↔ ∃ v, w ≺^[0] v ∧ Satisfies M v p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | constructor | case succ
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
⊢ Satisfies M w (◇^[(n + 1)]p) ↔ ∃ v, w ≺^[n + 1] v ∧ Satisfies M v p | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
⊢ Satisfies M w (◇^[(n + 1)]p) → ∃ v, w ≺^[n + 1] v ∧ Satisfies M v p
case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | . intro h;
obtain ⟨v, hwv, hv⟩ := by simpa using h;
obtain ⟨x, hvx, hx⟩ := ih.mp hv;
existsi x;
constructor;
. existsi v; simp_all;
. simpa; | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
⊢ Satisfies M w (◇^[(n + 1)]p) → ∃ v, w ≺^[n + 1] v ∧ Satisfies M v p
case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w ... | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
⊢ (∃ v, w ≺^[n + 1] v ∧ Satisfies M v p) → Satisfies M w (◇^[(n + 1)]p) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | . simp;
intro x y hwy hyx hx;
existsi y;
constructor;
. simpa;
. apply ih.mpr;
existsi x;
simp_all; | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
⊢ (∃ v, w ≺^[n + 1] v ∧ Satisfies M v p) → Satisfies M w (◇^[(n + 1)]p) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | intro h | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
⊢ Satisfies M w (◇^[(n + 1)]p) → ∃ v, w ≺^[n + 1] v ∧ Satisfies M v p | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
⊢ ∃ v, w ≺^[n + 1] v ∧ Satisfies M v p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | obtain ⟨v, hwv, hv⟩ := by simpa using h; | case succ.mp
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
⊢ ∃ v, w ≺^[n + 1] v ∧ Satisfies M v p | case succ.mp.intro.intro
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
⊢ ∃ v, w ≺^[n + 1] v ∧ Satisfies M ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | obtain ⟨x, hvx, hx⟩ := ih.mp hv | case succ.mp.intro.intro
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
⊢ ∃ v, w ≺^[n + 1] v ∧ Satisfies M ... | case succ.mp.intro.intro.intro.intro
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World
hvx :... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | existsi x | case succ.mp.intro.intro.intro.intro
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World
hvx :... | case succ.mp.intro.intro.intro.intro
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World
hvx :... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | constructor | case succ.mp.intro.intro.intro.intro
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World
hvx :... | case succ.mp.intro.intro.intro.intro.left
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | . existsi v; simp_all; | case succ.mp.intro.intro.intro.intro.left
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World
... | case succ.mp.intro.intro.intro.intro.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | . simpa; | case succ.mp.intro.intro.intro.intro.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | simpa using h | α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
⊢ ?m.54398 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | existsi v | case succ.mp.intro.intro.intro.intro.left
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World
... | case succ.mp.intro.intro.intro.intro.left
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | simp_all | case succ.mp.intro.intro.intro.intro.left
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World
... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | simpa | case succ.mp.intro.intro.intro.intro.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
h : Satisfies M w (◇^[(n + 1)]p)
v : M.Frame.World
hwv : w ≺ v
hv : Satisfies M v ((UnaryModalOperator.mop false)^[n] p)
x : M.Frame.World... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | simp | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
⊢ (∃ v, w ≺^[n + 1] v ∧ Satisfies M v p) → Satisfies M w (◇^[(n + 1)]p) | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
⊢ ∀ (x x_1 : M.Frame.World),
w ≺ x_1 →
RelItr (fun x x_2 => x ≺ x_2) n x_1 x →
Satisfies M x p → ∃ w', w ≺ w' ∧ Satisfies M w' ((UnaryModalOperator.m... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | intro x y hwy hyx hx | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
⊢ ∀ (x x_1 : M.Frame.World),
w ≺ x_1 →
RelItr (fun x x_2 => x ≺ x_2) n x_1 x →
Satisfies M x p → ∃ w', w ≺ w' ∧ Satisfies M w' ((UnaryModalOperator.m... | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ ∃ w', w ≺ w' ∧ Satisfies M w' ((UnaryModalOperator.mop false)^[n] p) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | existsi y | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ ∃ w', w ≺ w' ∧ Satisfies M w' ((UnaryModalOperator.mop false)^[n] p) | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ w ≺ y ∧ Satisfies M y ((UnaryModalOperator.mop false)^[n] p) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | constructor | case succ.mpr
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ w ≺ y ∧ Satisfies M y ((UnaryModalOperator.mop false)^[n] p) | case succ.mpr.left
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ w ≺ y
case succ.mpr.right
α : Type u_1
M : Model α
p q : Formula... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | . simpa; | case succ.mpr.left
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ w ≺ y
case succ.mpr.right
α : Type u_1
M : Model α
p q : Formula... | case succ.mpr.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ Satisfies M y ((UnaryModalOperator.mop false)^[n] p) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | . apply ih.mpr;
existsi x;
simp_all; | case succ.mpr.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ Satisfies M y ((UnaryModalOperator.mop false)^[n] p) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | simpa | case succ.mpr.left
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ w ≺ y | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | apply ih.mpr | case succ.mpr.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ Satisfies M y ((UnaryModalOperator.mop false)^[n] p) | case succ.mpr.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ ∃ v, y ≺^[n] v ∧ Satisfies M v p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | existsi x | case succ.mpr.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ ∃ v, y ≺^[n] v ∧ Satisfies M v p | case succ.mpr.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ y ≺^[n] x ∧ Satisfies M x p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Formula.Kripke.Satisfies.multidia_def | [151, 1] | [171, 18] | simp_all | case succ.mpr.right
α : Type u_1
M : Model α
p q : Formula α
n : ℕ
ih : ∀ {w : M.World}, Satisfies M w (◇^[n]p) ↔ ∃ v, w ≺^[n] v ∧ Satisfies M v p
w : M.World
x y : M.Frame.World
hwy : w ≺ y
hyx : RelItr (fun x x_1 => x ≺ x_1) n y x
hx : Satisfies M x p
⊢ y ≺^[n] x ∧ Satisfies M x p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.validOnAxiomSetFrameClass_axiom | [248, 1] | [248, 101] | intro F hF | α : Type u_1
Ax : AxiomSet α
p : Formula α
h : p ∈ Ax
⊢ 𝔽(Ax) ⊧ p | α : Type u_1
Ax : AxiomSet α
p : Formula α
h : p ∈ Ax
F : Frame' α
hF : F ∈ 𝔽(Ax)
⊢ F ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.validOnAxiomSetFrameClass_axiom | [248, 1] | [248, 101] | exact hF.realize h | α : Type u_1
Ax : AxiomSet α
p : Formula α
h : p ∈ Ax
F : Frame' α
hF : F ∈ 𝔽(Ax)
⊢ F ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.iff_definability_memAxiomSetFrameClass | [268, 1] | [269, 30] | apply definability.defines | α : Type u_1
Ax : AxiomSet α
P : FrameProperty
definability : Definability Ax P
⊢ ∀ {F : Frame' α}, F ∈ 𝔽(Ax) ↔ P F | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/Semantics.lean | LO.Modal.Standard.Kripke.iff_finiteDefinability_memFiniteFrameClass | [290, 1] | [291, 34] | apply definability.fin_defines | α : Type u_1
Ax : AxiomSet α
P : FiniteFrameProperty
definability : FiniteDefinability Ax P
⊢ ∀ {F : FiniteFrame' α}, 𝔽ꟳ(Ax) F ↔ P F | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Semiterm.substs_varSumInL | [16, 1] | [17, 111] | simp [varSumInL, Matrix.vecAppend_eq_ite] | L : Language
μ : Type u_1
inst✝ : Semiformula.Operator.Eq L
k n : ℕ
w₁ w₂ : Fin k → Semiterm L μ n
i : Fin k
⊢ (Rew.substs (Matrix.vecAppend ⋯ w₁ w₂)) (varSumInL i) = w₁ i | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Semiterm.substs_varSumInR | [19, 1] | [20, 111] | simp [varSumInR, Matrix.vecAppend_eq_ite] | L : Language
μ : Type u_1
inst✝ : Semiformula.Operator.Eq L
k n : ℕ
w₁ w₂ : Fin k → Semiterm L μ n
i : Fin k
⊢ (Rew.substs (Matrix.vecAppend ⋯ w₁ w₂)) (varSumInR i) = w₂ i | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Semiterm.emb_varSumInL | [22, 1] | [23, 95] | simp [varSumInL] | L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
k : ℕ
o : Type u_2
inst✝ : IsEmpty o
i : Fin k
⊢ Rew.emb (varSumInL i) = varSumInL i | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Semiterm.emb_varSumInR | [25, 1] | [26, 95] | simp [varSumInR] | L : Language
μ : Type u_1
inst✝¹ : Semiformula.Operator.Eq L
k : ℕ
o : Type u_2
inst✝ : IsEmpty o
i : Fin k
⊢ Rew.emb (varSumInR i) = varSumInR i | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | intro σ h | L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
⊢ ∀ ⦃f : Sentence L⦄, f ∈ 𝐄𝐐 → inst✝¹.toStruc ⊧ f | L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
σ : Sentence L
h : σ ∈ 𝐄𝐐
⊢ inst✝¹.toStruc ⊧ σ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | cases h <;> simp [models_def, Semiformula.vecEq, Semiterm.val_func] | L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
σ : Sentence L
h : σ ∈ 𝐄𝐐
⊢ inst✝¹.toStruc ⊧ σ | case funcExt
L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
f✝ : L.Func k✝
⊢ ∀ (e : Fin (k✝ + k✝) → M),
(∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) =
Semiterm.val inst✝¹... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | case relExt r =>
simp [Semiformula.eval_rel]; intro e h; simp [congr_arg (rel r) (funext h)] | L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
r : L.Rel k✝
⊢ ∀ (e : Fin (k✝ + k✝) → M),
(∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) =
Semiterm.val inst✝¹ e Empty.elim (... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | intro e h | case funcExt
L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
f✝ : L.Func k✝
⊢ ∀ (e : Fin (k✝ + k✝) → M),
(∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) =
Semiterm.val inst✝¹... | case funcExt
L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
f✝ : L.Func k✝
e : Fin (k✝ + k✝) → M
h :
∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) = Semiterm.val inst✝¹ e Empty.elim (Semit... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | congr | case funcExt
L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
f✝ : L.Func k✝
e : Fin (k✝ + k✝) → M
h :
∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) = Semiterm.val inst✝¹ e Empty.elim (Semit... | case funcExt.e_a
L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
f✝ : L.Func k✝
e : Fin (k✝ + k✝) → M
h :
∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) = Semiterm.val inst✝¹ e Empty.elim (S... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | funext i | case funcExt.e_a
L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
f✝ : L.Func k✝
e : Fin (k✝ + k✝) → M
h :
∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) = Semiterm.val inst✝¹ e Empty.elim (S... | case funcExt.e_a.h
L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
f✝ : L.Func k✝
e : Fin (k✝ + k✝) → M
h :
∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) = Semiterm.val inst✝¹ e Empty.elim ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | exact h i | case funcExt.e_a.h
L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
f✝ : L.Func k✝
e : Fin (k✝ + k✝) → M
h :
∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) = Semiterm.val inst✝¹ e Empty.elim ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | simp [Semiformula.eval_rel] | L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
r : L.Rel k✝
⊢ ∀ (e : Fin (k✝ + k✝) → M),
(∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) =
Semiterm.val inst✝¹ e Empty.elim (... | L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
r : L.Rel k✝
⊢ ∀ (e : Fin (k✝ + k✝) → M),
(∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) =
Semiterm.val inst✝¹ e Empty.elim (... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | intro e h | L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
r : L.Rel k✝
⊢ ∀ (e : Fin (k✝ + k✝) → M),
(∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) =
Semiterm.val inst✝¹ e Empty.elim (... | L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
r : L.Rel k✝
e : Fin (k✝ + k✝) → M
h :
∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) = Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInR i... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.models_eqAxiom | [83, 1] | [88, 81] | simp [congr_arg (rel r) (funext h)] | L : Language
μ : Type u_1
inst✝³ : Semiformula.Operator.Eq L
M : Type u
inst✝² : Nonempty M
inst✝¹ : Structure L M
inst✝ : Structure.Eq L M
k✝ : ℕ
r : L.Rel k✝
e : Fin (k✝ + k✝) → M
h :
∀ (i : Fin k✝),
Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInL i) = Semiterm.val inst✝¹ e Empty.elim (Semiterm.varSumInR i... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_refl | [104, 1] | [107, 15] | have : M ⊧ₘ “∀ x, x = x” := H.realize (Theory.eqAxiom.refl (L := L)) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
⊢ eqv L a a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : M ⊧ₘ (“∀' #0 = #0”)
⊢ eqv L a a |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_refl | [104, 1] | [107, 15] | simp [models_def] at this | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : M ⊧ₘ (“∀' #0 = #0”)
⊢ eqv L a a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : ∀ (x : M), op(=).val ![x, x]
⊢ eqv L a a |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_refl | [104, 1] | [107, 15] | exact this a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a : M
this : ∀ (x : M), op(=).val ![x, x]
⊢ eqv L a a | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_symm | [109, 1] | [112, 17] | have : M ⊧ₘ “∀ x y, x = y → y = x” := H.realize (Theory.eqAxiom.symm (L := L)) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
⊢ eqv L a b → eqv L b a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
this : M ⊧ₘ (“∀' ∀' (#1 = #0 → #0 = #1)”)
⊢ eqv L a b → eqv L b a |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_symm | [109, 1] | [112, 17] | simp [models_def] at this | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
this : M ⊧ₘ (“∀' ∀' (#1 = #0 → #0 = #1)”)
⊢ eqv L a b → eqv L b a | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
this : ∀ (x x_1 : M), op(=).val ![x, x_1] → op(=).val ![x_1, x]
⊢ eqv L a b → eqv L b a |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_symm | [109, 1] | [112, 17] | exact this a b | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b : M
this : ∀ (x x_1 : M), op(=).val ![x, x_1] → op(=).val ![x_1, x]
⊢ eqv L a b → eqv L b a | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_trans | [114, 1] | [117, 19] | have : M ⊧ₘ “∀ x y z, x = y → y = z → x = z” := H.realize (Theory.eqAxiom.trans (L := L)) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
⊢ eqv L a b → eqv L b c → eqv L a c | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
this : M ⊧ₘ (“∀' ∀' ∀' (#2 = #1 → (#1 = #0 → #2 = #0))”)
⊢ eqv L a b → eqv L b c → eqv L a c |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_trans | [114, 1] | [117, 19] | simp [models_def] at this | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
this : M ⊧ₘ (“∀' ∀' ∀' (#2 = #1 → (#1 = #0 → #2 = #0))”)
⊢ eqv L a b → eqv L b c → eqv L a c | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
this : ∀ (x x_1 x_2 : M), op(=).val ![x, x_1] → op(=).val ![x_1, x_2] → op(=).val ![x, x_2]
⊢ eqv L a b → eqv L b c → eqv L a c |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_trans | [114, 1] | [117, 19] | exact this a b c | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
a b c : M
this : ∀ (x x_1 x_2 : M), op(=).val ![x, x_1] → op(=).val ![x_1, x_2] → op(=).val ![x, x_2]
⊢ eqv L a b → eqv L b c → eqv L a c | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_funcExt | [119, 1] | [124, 128] | have : M ⊧ₘ ∀* (Semiformula.vecEq varSumInL varSumInR ⟶ op(=).operator ![Semiterm.func f varSumInL, Semiterm.func f varSumInR]) :=
H.realize (eqAxiom.funcExt f (L := L)) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ eqv L (func f v) (func f w) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this : M ⊧ₘ ∀* (“(!(vecEq varSumInL varSumInR) → !!(Semiterm.func f varSumInL) = !!(Semiterm.func f varSumInR))”)
⊢ eqv L (func f v... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_funcExt | [119, 1] | [124, 128] | simp [varSumInL, varSumInR, models_def, vecEq, Semiterm.val_func] at this | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this : M ⊧ₘ ∀* (“(!(vecEq varSumInL varSumInR) → !!(Semiterm.func f varSumInL) = !!(Semiterm.func f varSumInR))”)
⊢ eqv L (func f v... | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this :
∀ (e : Fin (k + k) → M),
(∀ (i : Fin k), op(=).val ![e (Fin.castLE ⋯ i), e (Fin.natAdd k i)]) →
op(=).val ![func... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_funcExt | [119, 1] | [124, 128] | simpa [Matrix.vecAppend_eq_ite] using this (Matrix.vecAppend rfl v w) (fun i => by simpa [Matrix.vecAppend_eq_ite] using h i) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this :
∀ (e : Fin (k + k) → M),
(∀ (i : Fin k), op(=).val ![e (Fin.castLE ⋯ i), e (Fin.natAdd k i)]) →
op(=).val ![func... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_funcExt | [119, 1] | [124, 128] | simpa [Matrix.vecAppend_eq_ite] using h i | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
f : L.Func k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this :
∀ (e : Fin (k + k) → M),
(∀ (i : Fin k), op(=).val ![e (Fin.castLE ⋯ i), e (Fin.natAdd k i)]) →
op(=).val ![func... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Basic/Eq.lean | LO.FirstOrder.Structure.Eq.eqv_relExt_aux | [126, 1] | [131, 138] | have : M ⊧ₘ ∀* (Semiformula.vecEq varSumInL varSumInR ⟶ Semiformula.rel r varSumInL ⟶ Semiformula.rel r varSumInR) :=
H.realize (eqAxiom.relExt r (L := L)) | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
⊢ rel r v → rel r w | L : Language
μ : Type u_1
inst✝² : Operator.Eq L
M : Type u
inst✝¹ : Nonempty M
inst✝ : Structure L M
H : M ⊧ₘ* 𝐄𝐐
k : ℕ
r : L.Rel k
v w : Fin k → M
h : ∀ (i : Fin k), eqv L (v i) (w i)
this : M ⊧ₘ ∀* (vecEq varSumInL varSumInR ⟶ Semiformula.rel r varSumInL ⟶ Semiformula.rel r varSumInR)
⊢ rel r v → rel r w |
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